Betweenness centrality in large complex networks

Abstract.We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent $\eta$. We find that for trees or networks with a small loop density $\eta = 2$ while a larger density of loops leads to $\eta < 2$. For scale-free networks characterized by an exponent $\gamma$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $\delta$. We show that this exponent $\delta$ must satisfy the exact bound $\delta\geq (\gamma + 1)/2$. If the scale free network is a tree, then we have the equality $\delta = (\gamma + 1)/2$.